# Randomized Block Design - PowerPoint

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Randomized Block Design

In chapter on paired t-tests, we learned to
“match” subjects on variables that:
 influence performance
 but are not of interest.

Matching gives a more sensitive test of H0
because it removes sources of variance that
inflate 2.
Lecture 17
2
Randomized Block Design (RBD)

In the analysis of variance, the matched
subjects design is called the Randomized
Block Design.
 subjects are first put into blocks
 a block is a group matched on some variable
 subjects in a block are then randomly assigned
to treatments
 for p treatments, you need p subjects per block

Lecture 17
3
Sums of squares in the RBD

We compute SSTreat as before. Compute
SSB (SS for Blocks) analogously:
 Compute deviations of block means from grand
mean.
 Square deviations, then add them up.

SSTotal is composed of SSTreat + SSE.
Does SSB come from SSTreat or from SSE?
Lecture 17
4
Where does SSB come from?

SST

SSTotal                       SSB

SSE
Residual
SSE
Lecture 17
5
Conceptual Formulas

SST = Σb(XTi – XG)2                    p-1
SSB = Σp(XBi – XG)2                    b-1
SSTotal = Σ(Xi – XG)2                  n-1
SSE = SSTotal – SST – SSB              (b-1)(p-1)
= n-b-p+1
MST = SST/(p-1)
MSB = SSB/(b-1)
MSE = SSE/(b-1)(p-1) = SSE/(n-b-p+1)

Lecture 17
6
Summary table

Source      df     SS                MS          F

Treat        p-1   SSTreat           SST/(p-1)     MST/MSE
Blocks       b-1   SSB               SSB/(b-1)     MSB/MSE
Error    n-p-b+1   SSE               SSE/(n-b-p+1)
Total        n-1   SSTotal

Lecture 17
7
Computational Formulas

CM = (ΣX)2             SSTotal = ΣX2 – CM
n
SSTreat = ΣT2i – CM    SSB = ΣB2i – CM
b                    p
SSE = SSTotal – SST – SSB

p=# of samples   Ti=Total for ith treatment
b=# of blocks    Bi=Total for ith block

Lecture 17
8
Randomized Block Design – Example 1a

H0: 1 = 2 = 3
HA: At least two differ significantly

Statistical test:        F=           MST
MSE

Rej. region:             Fobt > F(2, 8, .05) = 4.46

Lecture 17
9
Randomized Block Design – Example 1a

CM = 104834.4

SSTotal   = ΣX2 – CM
= 782 + 812 + … + 942 – 104834.4
= 105198 – 104834.4
= 363.6

Lecture 17
10
Randomized Block Design – Example 1a

SSTreat = Σ(Ti2) – CM
b
= 4012 + 4212 + 4322 – 104834.4
5       5    5
= 104933.2 – 104834.4
= 98.8

Lecture 17
11
Randomized Block Design – Example 1a

SSB = ΣB2i – CM
p

SSB = 2442 + … + 2712 – 104834.4
3          3
= 105075.33 – 104834.4
= 240.93

Lecture 17
12
Randomized Block Design – Example 1a

SSE = SSTotal – SSTreat – SSB

= 363.6 – 98.8 – 240.93

= 23.87

Lecture 17
13
Randomized Block Design – Example 1a

Source     df    SS                MS      F

Treat      2      98.8             49.4    16.55
Blocks     4     240.93            60.23   20.18
Error      8      23.87             2.98
Total      14    363.6

Decision: Reject HO – average scores do differ
across exams.
Lecture 17
14
Randomized Block Design – Example 1b

H0: w = A
HA: W ≠ A

(Note: this is a post-hoc test. We’ll do N-K.)

Statistical test:   Q = Xi – Xj
√MSE/n

Lecture 17
15
Randomized Block Design – Example 1b

Rank order sample means:

Edison        Wilbur          Orville             Thomas   Alva
90.3          86              81.3                80.6    79.67

r=4

Qcrit = Q(4, 8, .05) = 4.53

Lecture 17
16
Randomized Block Design – Example 1b

Qobt:

86 – 79.67   =            6.33    = 6.35
√(2.984)/3                0.997

Reject HO. Wilbur & Alva differ significantly
on their overall average on the 3 exams.

Lecture 17
17
Randomized Block Design – Example 2a

H0: 1 = 2 = 3
HA: At least two differ significantly

Statistical test:        F=           MST
MSE

Rej. region:             Fobt > F(2, 8, .05) = 4.46

Lecture 17
18
Randomized Block Design – Example 2a

CM = 35072      = 819936.6
15

SSTotal = ΣX2 – CM
= 2102 + 2452 + … + 2902 – 819936.6
= 855701 – 819936.6
= 35764.4

Lecture 17
19
Randomized Block Design – Example 2a

SSTreat = Σ(Ti2) – CM
b
= 12282 + 11872 + 10922 – 819936.6
5        5     5
= 821883.4 – 819936.6
= 1946.8

Lecture 17
20
Randomized Block Design – Example 1a

SSB = ΣB2i – CM
p

SSB = 6042 + 7272 … + 9582 – 819936.6
3      3        3
= 852973.67 – 819936.6
= 33037.07

Lecture 17
21
Randomized Block Design – Example 1a

SSE = SSTotal – SSTreat – SSB

= 35764.4 – 1946.8 – 33037.07

= 780.5

Lecture 17
22
Randomized Block Design – Example 1a

Source     df    SS                MS   F

Treat      2     1946.8   973.4         9.977
Blocks     4     33037.07 8259.27       84.656
Error      8     780.5    97.563
Total      14    35764.4

Decision: Reject HO – weights do differ across the
3 time periods.
Lecture 17
23
Randomized Block Design – Example 1b

Lecture 17

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